 V nekaj trebih lektur. Vsega lektura dobro je zelo, da je zelo, da je zelo, da je zelo. Neliko nismo tudi nekaj, da je v USA, vsega je v Bostonu, in da se dobro v Bostonu dobro. Zato smo vsega izgledali na nekaj prezentacij, prezentacije kvantumov, ki so pričo se zelo vsega vsega vsega vsega vsega vsega vsega vsega. Zelo vsega vsega vsega vsega in vsega vsega vsega vsega, So systems that you can. Think of a similarity to the Keep advice type of modeling in the context of others classical systems. Of course you do a quantum treatment it will be slightly different so you will have to do with matrices and okay this is something you will see je veliko zelo svobodno, veliko zelo svobodno, ali se vseč je vseč samo. Zato videljš, da se vseč zelo sega in zelo sega in tudi v kvantumih modelji zelo sega vseč vseč vseč vseč vseč vseč vseč vseč vseč vseč vseč. Vseč videljš vseč vseč vseč vseč vse je, da je zelo način, je zapravil v trajne, zelo v teželnih zelo, na tessin, kaj je. Vse je, da je vse izgleda, vse je, da je, da je, da je, da je, da je, da je, da je, da je, da je, da je, v kratom verju. Zato to je vse objev, zato način... Vsih je mnoj različ, če je zelo in najboljših vsega vzela in vsega vzela, da sem tudi vsega. To je produk na moj interaktivne... Vsega vsega vzela... Ivan Latela ki je tudi v Barcelonau. Zdaj smo začali kolaboracijo 6-7 rov. Zdaj to je nekaj nekaj kolaboracijo, tako, da je vzvečal v zelo, v zelo, da je vzvečal vzvečal vzvečal vzvečal vzvečal vzvečal vzvečal vzvečal vzvečal. Počučali smo se vzvečal, da smo zelo vzvečali o zelo, da smo vzvečali, da je vzvečal še je vzvečal vzvečal v zelo, z vzvečal vzvečal vzvečal. I ino nekaj, ovečal, če bo se vzvečal... OK. Zazvo so imamo vseč, očini vzvečal, je to, Prvno je to druga leksu, ali včetno ne bomo vsega leksu, ker je to pričo, da je bilo vsega leksu, tako sem spliti vsega leksu v 1a in 1b, je to vsega leksu, tako v 1a in 1b. In potem, okej, vsega leksu, This is a sixth lecture if you count properly the number of sessions that we have had. So, I'll go backward to the first lecture today. With concepts related to aditivity and long range interactions. But I will discuss these topics in a wider sense, in a thermodynamic sense. Čutko, zelo vši tudi, da bomo vzela vzela termodanamik. Vzela, da termodanamik je to vzela in je izvedne značno. Zato je tudi počkaj, da je vsečo informacije na sistem, in potem, da je vsečo, da je vsečo, da je vsečo, da je vsečo, da je vsečo, da je vsečo, da je vsečo, As in as, as you will see how it's possible to see the counterpart in stat mech. And so today will be devoted to the introduction of a new ensemble that maybe you never heard about. It's an ensemble where the number of particles fluctuates, the number of the energy fluctuates in volum je lažen. Tudi tudi tudi tudi lažen. Zdaj sem si, da je odvržal, da se zelo tudi tudi tudi lažen in tudi je vsega dobro. Zelo ne zelo tudi tudi lažen, ki se tudi tudi tudi lažen. In rezoj je, če pri svoj počet, that if the interactions, this was done by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by by. Long ago in the thirties that if the three quantities fluctuate all together, there are no equilibrium states. ta bilomče ne zelo učnično. Ne zelo je nekaj, da vse zelo vse začela, ta če je na štahla, nekaj nezelo je začela. Vse boče ne zelo vse začela, boče ne je začela. Taj je začelo učnične zelo. Zelo je zelo se vsega, zelo je začelo na tudi, ovo je nezelo, je začelo na zelo, zelo je začelo na zelo. Zelo je začelo na zelo. A termodynamic system. If the interactions are long range, there can be equilibrium states. Not only there can be equilibrium states I will show today a model that besides having equilibrium state, it has phase transitions and also phase transitions and phase diagrams are in equivalent in se prav ed ali oset. To je nov je gener� голoba, taآo se naj 말� Oke in sp auton in sağuj se, da sočil nam se izcušno poelo tje vone, nosimo in constranje na ne večske z coinšli iz UP. V pripad sac to zestaj vstavum odz fewerc There are profound things, I think about this aspect that the long range interactions allow this new ensemble. I would say that, as a suggestion to understand why this ensemble exist in long range interactions, you can think that usually what you have to do when you have a gas, you have to put a gas in a volume. In vseč, da jih se vseče, da se vseče. Vseče, da se vseče in energije. Sistemovati in vseče in vseče in vseče in vseče in vseče. zelo počkaj je zelo počkaj, ki ne počkaj vzela izgledaj. Vzelo počkaj je zelo prijevno v vse. To je nekaj ideja. Zelo je to izgledaj, da se počkaj počkaj počkaj ne vse odgledaj. Ok, zelo to je več intuitujivo, je zelo, da se taj vse, je vse objezena bilo, taj nekaj terasti. A potem, zelo, da selamo dolje stručno vzelo do modu, da je zelo vsevajal V70-i, z Valter Tilling. In, da je zelo, ta ljubov, zelo, da sem počutila v slaku, nekaj pačen počutih, ker sem nekaj počutil, da je problem, kaj je zelo, v 70-jih, in je problem nekaj negativne spesivne kete, in je korekli, da se počuti, zelo, da je problem v kontekstu, zelo, in vzelo, in zelo, in zelo, in zelo, in vzelo, vzelo, in vzelo, ki se zelo, da je zelo, in vse in zelo, kanoniko in mikrokanoniko in in in in in in in in in constreniko in zelo. In so, so je to več semplj modern, in je to, da je to, da je to, da je to, da je to, da je to informacija. In dan, in dan, In počim bomo izgledati, kako vso nerevnačne simulacije v tem nekaj nesemboljo, načo je montekarlo, ovo je radišljeni na toh nekaj nesemboljo. To je zrp. Zabijam tudi, kako najčešte, kako izgleda na 2015. Začnem način. As you know, as you have known, the additivity is a basic property of systems with short range interactions. It implies that energy. Let us look at additivity in a different way. Instead of looking at additivity as a property of the energy itself, let us put it in relation with the homogeneity of thermodynamic potentials. This is another way to look at additivity. In this new sense, additivity implies that energy is a linear homogenous function of entropy, volume and the number of particles. By the way, the property that you use when you derive the thermodynamic relations that imply energy, entropy, volume and number of particles. For instance, the way you write the work as in the so-called TDS equations, I used to call them in this way, but I will repeat this scaling property in a while so you will see where I use the additivity. As we have seen, the systems with power-low decaying interactions are non-additive. If the power alpha is smaller or equal than the dimension d of the embedding space. In fact, we have not seen many cases in which alpha equals d. It is a sort of limiting case. It is an interesting one, for instance, dipolar systems. There are many examples in physics, self-gravitating systems, Coulomb and dipolar systems, but also two-dimensional fluids like the atmosphere. There are plenty of applications in this field of the tool that we have learned. The non-additive interaction lead to ensemble in equivalence, which implies negative specific heat, temperature jumps, broken ego DCVs. There are plenty of new phenomena, quasi-stationary states, plenty of new phenomena that arise in this field. Just to put it in context, let us think how Gibbs introduced the so-called grand canonical ensemble. The idea of Gibbs is that it costs energy to change the number of particles. So you have a system and Gibbs thought of copies of the system, several copies of the system that are distinguishable, but they can have different number of particles. And having or not having a certain number of particles costs energy. OK, now let us generalize Gibbs idea. Now you have copies of a system that is long range. Each of the system is long range. And it can have a certain number of particles, a certain energy, a certain entropy. And you are making copies replicas of this system. And each of the system has n particles, energy, entropy s, and volume v. What changes in this game? Julio, could you please switch off your mic? So this was my glasses. Julio, Tani, Raffelli. OK, thank you. Switch off everything. So the only thing that varies is the number of copies. So the total energy will be the number of copies times the energy of a single system. The total entropy will be the sum. There is no interaction among the copies. It is like in the Gibbs construction. There is no interaction between copies of the grand canonical. So the construction is very similar, in fact, to the construction of Gibbs. So the total volume will be the total number of electrons times the volume of the single system. The total number of particles will be the number of volume. So now, OK, so this is very much inspired. I advise you to read the work of this guy, Hila. So the idea of Hila was different, but is very much connected with this one. So instead of creating copies of the system, he subdivides a single system, which is interacting in parts. And in fact, the name that he gives to this energy is the subdivision potential. So you can write a thermodynamic relation for the total energy, which includes the variation in the number of copies. OK, you see that it comes after the Gibbs term. So the variation of the energy of the total energy might be due to a variation of the entropy, TDS, to a variation of the volume minus PdV. For historical reason, there is a minus here. It depends on the sign of the work. Then the Gibbs variation, which is the chemical potential times the variation of the number of particles in a single system, plus a term, which takes into account the number of copies that might cost energy. OK? So we will see in a very simple way that in the case in which each single system is additive, the cost of adding a copy is zero. While if each single system is long range, is a bit magic, if each simple system is non additive, then it will cost energy to add the copy. So let's do it. So the proof is two lines. So you change the number of copies by factor xi. In such a way that the entropy is divided by factor xi, the volume is divided by factor xi, and the number is divided by factor xi. And is n total. OK, so is the n total. You do this on the total quantities. Then, of course, you just apply these relations, and you see that if you do like this, the variation of the entropy of the volume and of the number of particles is zero. And this implies that if this is zero, or this is zero on the right hand side, there is a direct relation between the variation of the total energy and the variation and the replica energy epsilon. Capital E. OK. But in additive systems, the energy is a linear homogeneous function of the entropy. So the energy of the system two is nothing but the energy of the c. So you see that I have to divide by factor xi. And therefore, dT is zero, which implies that E must be zero. So if the energy is a linear homogeneous function of its arguments, the replica energy is zero. But we know that for long range systems this is not true. So there must be an energy associated with the fact that you add a copy to the system. Let me move my. OK. So now you have to work a little bit with thermodynamics. So this is the integration of the equation for fixed single system properties. So I fix the numbers in the single systems. And just one step, you prove this relation. So if you divide by the number n of systems, you get this explicit relation that relates the energy, the entropy, the pressure and the volume and the replica energy. You have to do it to be convinced that it is OK. OK. So I put all the ingredients to do it. Then you differentiate this relation. And you take into account, go and look in a book of thermodynamics to the first TDS relation, to the first TDS equation. And you get an equation for the variation of the replica energy. OK. This is a relation that appears in thermodynamics book. It's called the Gibbs-Duhem relation. And you see that usually this Gibbs-Duhem relation has a zero on the left side, which implies that you cannot fix temperature, pressure and chemical potential independently. They are related by an equation of states in additive system. But here, because of long range interaction, you have an extra term, which is due to the variation of the energy associated to the variation of the number of copies. So the Gibbs-Duhem relation is violated. So this is one news. The Gibbs-Duhem relation, which usually takes for granted in thermodynamics book, is violated, but simply because in thermodynamics, you assume that the system is additive. If the system is non-additive, the Gibbs-Duhem can be violated. Moreover, if you integrate this equation, you can get an equation for E. So if you are able to compute, sorry, for calligraphic key for the replica energy, so if you are able to compute in a given ensemble for a given model, you are able to compute the thermodynamic functions and the equation of state, then you can write explicitly what is E. So it's not mysterious, this E, you can compute it. So the first thing that we did, and I will not repeat the historical steps, was to compute this E, this calligraphic E, in several different cases. So take a system in beam field and with decaying interaction, compute E. Or take the tilling model and compute E, for instance, in the microcanonical ensemble, since the model was solved in the microcanonical ensemble. So we took some time to familiarize, maybe you will see in the papers, but I don't want to follow the historical thread because it will be too long. So the procedure will be slightly different, more logical. So after we convinced ourselves that this E was calculable on some models directly, we started to think in a different way. And this different way is, since the kibs needed a new ensemble to formulate the problem of changing number of particles, there should be an ensemble that takes into account the variation in the number of copies. And this ensemble is this one, it's very simple to write, in fact, is a probability distribution, where you have three Lagrange multipliers, one for the number, one for the energy, and one for the volume. So there is no mystery in that. And then the corresponding partition sum is a partition sum where you have to, here the energy is discrete, but you can replace the sum over the energy states by an integral over the energy, of course you can replace the sum over the volume, everything in the continuous limit will be replaced by this. And so then the change in average energy can be written, you know that the expression has a change in the probabilities, in the usual way that you do for ensembles. Ja, so the index i runs over the copies on the replicas, right? The index i is the energy states. You don't see here the replicas, you see the single state, ok? You see the single, this is the probability distribution of the, ok. So this can vary, ok, the probability distribution of the single state and the variation induces a variation in the energy. So now I rewrite the variation in the energy using the conservation of probability and I obtain a relation, this relation where and bar v bar are the average number and the volume. And this is to be compared with this thermodynamic relation, ok? Which now allows to identify the thermodynamic functions so you can write the temperature in terms of the multiplier beta, the pressure, in terms of the multiplier gamma and beta and the chemical potential in terms of the two multipliers. And this is the entropy associated with this construction. So then substituting this relation in the expression for the probability in the entropy you get the expression of the replica energy. This is where the replica energy enters, ok? So from the thermodynamic relation for the replica energy and having computed these averages and the entropy I can obtain the replica energy which is nothing but as usual for the other ensemble which is nothing but kbt times the logarithm of the partition sum for this ensemble, ok? And by differentiation one obtains the relation for the replica energy so you can see that in this ensemble the entropy is minus the derivative of the replica energy with respect to temperature at fixed pressure and chemical potential the volume is the derivative of the energy of the replica energy with respect to pressure at fixed temperature and chemical potential and the number of particles you fix the other two. So you fix two over three and you get the fluctuating the average fluctuating quantities, ok? So this is the construction it is, ok? And of course there are legend transforms to the other ensembles so I can write the partition sum in one constrained ensemble for instance in terms of the partition sum in the ensemble in which temperature, volume and number of particles are fixed by simply legend transforming with respect to this would be the canonical ensemble, no? This is the canonical ensemble by legend transform with respect to the number of particles and with respect to the volume and obtain the partition sum in the unconstrained ensemble ok, this would be the canonical partition sum but I could also obtain it from the grand canonical partition sum which is this horrible symbol here by legend transform by laplace transforming with respect to by laplace transforming with respect to the pair, pressure and volume, ok? So I have several ways of deriving this new partition sum and this is a graph where I show all the ensembles that we could construct by this type of construction so I start for the entropy with the entropy which is the thermodynamic potential in the micro canonical ensemble where the energy, the volume and the number of particles are fixed, ok? Now I can let the energy fluctuate and I get the massier potential or the free energy if you want in the canonical ensemble in this construction as you have seen I am losing the invertibility of the Legend-Fenzhel transformer so there can be response functions that have not the right sign and in this case the response function which can be negative in the micro canonical ensemble but the more the yapmış polishive invariant the more specific it is which is the response function with respect to the energy so the variation of the temperature the variation of the energy with respect to the temperature the inverse of the variation of the temperature with respect to the energy in the micro canonical ensemble while it's variation of energy izgledajte vse rešvene v te dve odgledaj. V ena odgledaj je tudi vse poslutne. Tudi je to izgleda, kaj je tudi saj vse zelo. Zelo je to izgleda, da je vse izgleda, da je izgleda, da je izgleda, zato je zelo stabilitave, da se izgleda, in nekaj zaštej entropi, in tudi smo očetili tako, ok? Dakle smo očetili tvoj problem, da smo videli mnogo modelov, in da smo vedeli toga način, in bojte je tobe v tem, ok, no, there are two possible ways, Either, you free the volume, OK. Ida jih vse bolj, in jih jih vse prešel ensambolj, ko je to, ko je ensambolj, ko je komis, ko jih jih vse bolj, da je fizicist, ker vse vse način je konstant atmosferične prešelje, OK, zato... In ensambolj, ko je fizicist, to je najbolj vse, fizicist kaj je z komisem, z komisem. Zato se vzicega vzicega vzicega vzicega, vzicega vzicega vzicega. Zato vzicega vzicega vzicega. I je inovojnje obrženje, ko je kompresibilit, ki je nekaj negativ vzicega kanonima, in to, da je vsečo pozitiv na izobariče, v izobariče assembling. Kako ti različi, in je ideti v to, da bomo so vzlizili na izobariče in ovorili, kjer lahko tega vsečje. To je na začinaj izobariče ensambo, je mikro kanonika. Vseč je vsečč, energij, voliume in delenje partijov. Na izobariče, Or, you go to the grand canonical, where you let the number of particles fluctuate, and you have another associated, now I don't remember the name of this response function, but you have another associated the response function in which you can, that is necessarily positive in the canonical ensemble and can be negative in the grand canonical ensemble. Is the response associated with the fluctuations of the number of particles. Now, the two lines join in this new ensemble, so if you let in the grand canonical fluctuate the volume, then you reach the unconstrained, or the alternative, you let the number fluctuate in the isobaric, you let the number fluctuate in the isobaric, and you reach the unconstrained, which for massier potential has better times the, it was a bit lengthy, but this is the panorama that we have in front. So, and all types of inequivalences could be checked at any of these steps. So, there are models in which you can check inequivalence at any step when you release a quantity that is fixed in one ensemble. So, today we will concentrate on the new inequivalences that arise because of the existence of this new ensemble. So, the inequivalence, for instance, of the unconstrained ensemble with the isobaric ensemble. And in order to do that, I need the model, which is adapted for fluids. No, yes, so these inequivalences are always associated with the first order, first transitions in all these cases. Yes, yes, you need, so, yes, in general, yes. I would say that this general feature is that the associated thermodynamic function should have a region of where it is with a supporting line, with a supporting line. So, there should be an unconcavity of one thermodynamic potential and this can appear only at first order phase transition, not at second order phase transition. Do you have coexistence essentially? Yeah, you must have coexistence, a region of coexistence of the two phases, yes. I have no general proof of that, but this is what is, should happen in all these steps, yes. And also in the example that I will show today, it is like that. Okay, so, I think now it is clear and it is even very general what I have shown now. You see, general, so, if you have questions on that because I will now go to the model. No, everything clear? Yeah, it's a bit shocking the fact that you have this new thing. So, even for us it was not easy to accept that we had confinced ourselves. Okay. No, I was just wondering because when we spoke previously about replicas in all other courses, where when we were averaging over disorder and we would take the copies of the system with different realizations of disorder. So, I was wondering if this is like the underlying mechanism behind this, or this has nothing to do. No, no, we didn't have a good name for these copies. We call them replicas, Gibbs called them copies. So, we decided to call them replicas, but they have nothing to do with the problem in random systems. These are distinguishable copies of the same systems. This is the definition of a replica. Okay. Okay. So, now, let us go to a model that I like a lot. It was proposed by Walter Thiering. And in a paper that appears, and I have the copy that I will put on the web, is a certain preprint, and I like very much this paper. It's very well written. And it starts by asking how can be that specific heat is negative, and then, okay, it's a bit lost in the beginning. So, it starts by trying to average the... So, the issue was raised by Lynden Bell and Wood in a paper in 1966. And there was no answer to this issue, and the statistical mechanics community was sort of skeptic about the fact that an astronomer was questioning the basics of statistical mechanics by saying that specific heat could be negative. So, they were a bit suspicious that this guy, although he was a famous astronomer, he became a professor in Cambridge and one of the most famous astronomers in these years, so maybe he was a little bit too arrogant to say things that statistical physics community didn't like. But Thiering had the right attitude. So, he tried to understand. So, in the first part of the paper, he tries to do a trick to deal with gravity, with Newtonian gravity. And the trick was sort of like the entropic construction that we've seen in the Lynden Bell paper. The trick was sort of averaging over boxes. So, he was taking the system and then was trying to have an idea what happens if you average over boxes, and he gets a formula at the end of his first paragraph where a very complicated formula that he can rewrite. So, his objective was, of course, to compute the entropy and his idea from the start is very clear that he was thinking to ensemble in equivalence. So, that the only way out of this dilemma put by Lynden Bell was understanding what happens for thermodynamic entropy, for micro canonical entropy. So, he was pointing to the inequivalence of ensembles from the start. But then in the second paragraph of the paper he says, okay, I have all these boxes, but what if I replace by just one box? So, and that's very simple. It's a fantastic idea because everything is clear and you compute it into lines. So, you have a system now, it's put in a box, okay. You need a box. At that time, of course, people wanted the box because with gases it was like that. So, it doesn't work any... Okay, yes. So, and you have kinetic energy and you have potential energy and the potential energy essentially counts the number of particles that are in the inner box. The number of particles that are in the inner box means that if you have a pair of particles that are in the V0, in the volume V0, which is contained in the volume V, smaller than the volume V, then if they are together, they interact and they attract each other with a strength minus nu, okay. So, this is the theta function, is the theta function of the volume V0 and if the position of the particle is in the volume V0 and also the other particle, the particle i and the particle j must be in the volume V0, then they attract with a potential phi. If one of the particles is in the volume V0 and the other is outside, it doesn't count any energy of interaction, they are free to move and if both are out, they don't interact, okay. So, then I will finish the lecture by showing the entropy and the free energy of this model, but okay, so the total potential energy in the large and limit is just minus nu and zero square. So, you just count the number of particles that are in volume V0. Okay, so that's the problem and you have to solve the micro canonical entropy. Now, I leave it to you because from the knowledge that you have of counting with the spin models that we have seen and so I give you just two steps. First you have to do the integration over the momenta, which is easy to do. So, you write the delta and you integrate over the momenta is a Gaussian integral so you can do it and then you are, you have to do the integral over the positions. Okay, but the integral over the positions it is very simple because you have to integrate over the position dq1, dqn over the particles and particles can be only in volume V0 and in volume V0 and in volume. So, essentially there will be, so, okay. So, you have to remember that you have to, this is, so, okay, you have to integrate some integrant after having integrated over the momenta and so the count, the relevant counting factor is n factorial divided by n0 factorial, which is the number of particles in the volume V0 and n minus n0 factorial. This is the relevant counting factor. Now, this n factorial will be killed by the Gibbs paradox factor that you put in the omega. So, you remain with that and then you have to maximize with respect to n0. So, you count all states. So, you have to count all states irrespective what is in the zero value. And there you use the saddle point. No, you have to maxima at large n. You have to, so, you use the steering approximation and you do the large n. It's really very simple. You get the results and these are the parameters as a tilling we're putting them. So, this is the number of particles, the ratio of the number of particles that are in the gas phase. So, 1 minus n0 over n. So, these are the number of particles that are in the gas phase. This is the rescaled energy. So, since the energy in this model is growing like n square and there has not been, yes? Yes, sure. Maybe it's a very stupid question, whether we're dividing by n0 or the interaction term by using the fact trick, because, I mean... In the previous slide, like in the definition of the Hamiltonian. Why we don't divide by n0 square here? No, no, by n0 in the definition of the Hamiltonian for the interaction term. Ah, yes, yes. So, this was the time in which they were not using the cut-series scaling. So, I do it naively as they were doing. Very good question. In the cut-scaling, so the system is fine at n for the moment. And in fact, if you do the thermodynamic limit without doing the cut-scaling, you run into troubles. And what Turing did was to define the energy in a different way. So, it defines the energy divided by n square and then it divides the temperature divided by n. And in this way, n is intensive, ok, and the energy also is intensive. And the reason of this plus one is that in this way all the energies are positive in the model. Ok, it's just a normalization. So, it's a different scaling, but it is equivalent to the cut-scaling in the end. Ok. It could have done introducing the cut-scaling Thank you for the question. It was a bit sloppy. And this parameter eta is very interesting because for some reason he likes this logarithm of this because in the previous calculation he had a ratio between the volumes that was exponential of some function. So then it calls the scaling factor for the volume. He calls it exponential of this, he was calling it f, but f makes confusion with the free energy so we decided to call it eta. But these are exactly the parameters that he used in this scale. Then, if you do this integral there, you get this entropy which is the the fraction of the number of gas of this expression. You see that again this entropy has some fermionic flavor through this term and this is the part coming from the kinetic energy. So this entropy does not have the decreasing part that we have seen in spin systems. It is an entropy of a gas essentially, of an interacting gas. So if you put more energy in the system the energy which is not taken by the potential energy will be taken by the kinetic energy. So the volume will keep increasing as you put more energy. So the entropy will never decrease. So it will always increase. It is a generic feature of a system with kinetic energy. It is the first one that we see in fact. So it is important to you could check for instance that if you define a micro canonical temperature using this expression for the entropy there is no negative temperature region. So this is even more clear because sometimes you could tend to associate the negative specific heat without negative temperature. This is a system without negative temperature. So all temperatures are positive for all the states. Nevertheless, the specific heat can be negative. And then you have canonical free energy which is as you see I have to divide the free energy and take the inf. So essentially this is the legend-fenchel transform of the canonical free energy. A plot, unfortunately I wanted to add a slide on this, but OK, the original slide, but I take it from a paper that we wrote recently. So we realize a couple of years ago that no one had computed the phase diagram of the model. So we decided to compute the phase diagram of the model and the situation of ensemble in equivalence that appears here is again associated with the first-order phase transition which is different in the two ensembles. So you see that in the OK, tau is the rescaled temperature. Tau is the rescaled temperature. So I'm plotting the phase diagram in the two variables that rescaled temperature and the parameter eta. And I can compute the critical point in the two ensembles by expanding, by using a Landau expansion in the order parameter which is the fraction of the volumes. So the order parameter is the density in this case. And if I use the variables adapted to the canonical so the intensive variable adapted to the canonical ensemble, I get a line of first-order phase transition with the end point which is here which is a second-order phase transition in the canonical ensemble. And if I do it in the micro-canonical exactly you remember the area that was at the form of a fish in the models that we have solved, spin models. In this case is this area here which are states that are not accessible in the micro-canonical ensemble. And in fact if you cross these, so for instance at fixed temperature you cross this area here there is a jump in etha. I remind you etha is essentially the log of the fraction of the volumes the external volume v0 is a parameter of the Hamiltonian you have to think of v0 as a parameter of the Hamiltonian while the volume is the volume containing the gas which is fixed. When I take the thermodynamic limit I let the volume increase with the number of particles that increase in such a way to have a constant density but v0 is a parameter of the Hamiltonian. By the way there is a very interesting limit that you could do is when you take the volume to infinity you could take the volume v0 to 0. And in this case because you go back to a short range interacting system. So v0 is a sort of finite in the in the larger volume limit. So it's affecting a finite part of the system and this also is one of the ingredient of the in equivalence in this in this model. If you want this is a modern way of looking what Tiring was looking. Tiring was always putting himself in this region so he had a jump in density and a jump in temperature and he was never looking at the full phase diagram. We decided to look at the full phase diagram and the full phase diagram is the following. So you have a critical point which is not the same in the two ensembles as for the Blumechappel model and for the they don't agree the two in this case is the critical point. So we have to solve the discussion at that point because coming also to the question of Matteo. So where I should expect ensemble in equivalence near first order phase transition. Now in this case the first order line is not the same you can see but there is no tri-critical point in this model so there is no region of second order and first order transitions. There is a line of first order transition in both ensembles that finishes in the critical point but the positions of the two lines and of the position of the critical point are not the same. So that creates the ensemble in equivalence. So the phase diagram is intrinsically different in this ensemble. Ok, so this is if you want the first part of the lecture I took one hour and in the second part I will show the application of the formalism of the unconstrained ensemble to the theory model. Now the problem that we had when we tried to apply the unconstrained to the theory model the theory model is not rich enough to create states in this new ensemble equilibrium states in this new ensemble. So we were very depressed because we were convinced that there were states in this new ensemble but we could not find them with the theory model. So you will see that the trick was to enrich the theory model to put if you want unphysical pressure from outside. So besides attracting the particle v0 you push the particle that are outside into the volume v0 then you get equilibrium states. So the model was not rich enough in parameters in order to give equilibrium states in this new ensemble. I think if you take other models you have to play with parameters in such a way that you get is not for given that you get a new ensemble. But we were expecting them and you have to modify the theory model in such a way to allow a richer. We were looking for phase transitions and I remember I gave a talk two years ago in Buenos Aires before the pandemic and I put the question to the audience we could find equilibrium states but we could not find phase transitions and then when we started to discuss these then the idea was to put the particle with an excluded volume. So the final step is to have a theory model with an external pressure plus a size for the particle. And if you put all them together then you get everything. Equilibrium points, phase transitions and ensemble in equivalence. So you have to sort of enrich the model. So just to see what do you mean is are you changing the Hamiltonian? Not changing the Hamiltonian. So by this pressure term how do you change the Hamiltonian? You will see. I add a term in the potential energy I will do it explicitly I add a term in the potential energy which introduces a pressure from outside. I have to change the Hamiltonian So if I study this model in the unconstrained ensemble I don't get equilibrium states. So if you want to get a picture it's like thinking of these particles inside an inflated balloon which lives in a system with pressure peak. Yes. Yes. I don't know how physically it can appear but okay. It's an unphysical modification. Without V0 this is an ideal gas, right? It's an ideal gas. It exerts pressure on or even say on a free piston. You could think also if one does a sort of repelling particle so in some sense the particle should attract when they are in V0 and should repel when they are in V1. This would be also analog. So they are not free particles? No. Even when they are in V1. So here they are free particles when they are in V1. In the T-ring model they are free particles. But when they are in V1 they repel. If they are in V0 or they repel if they are in V1. How to do this? I don't know because they are the same particles. Why they should repel in V1 and attract in V0. So one could think for instance of shining a laser in the inner part in order to change the sign of the interaction among the particles. Particles that naturally would repel if you shine a given region then they should attract. This would be the physical way of I have no idea. It's totally unphysical. But one has to think of a way of realizing this. I think in social interaction it's easy. So either you like each other or you don't like each other. If you are in the bad place you can hate each other. I think it's a non-physical system and it's easy to think. Just to finish I think the idea of tearing I find it very interesting and I think he is a pioneer in this solution of the problem. Although people attribute these to others like Lynden Bell and others. But I think this paper by tearing is really the corner point in the solution of the puzzle. And this model that I am doing in the last lecture should have been in the first lecture. But for logical reasons it's better to introduce somebody in equivalence in spin models that are models of magnetism that are more easy to solve in some sense and more familiar to us. But I think this model is a really fantastic way and is the solution of the puzzle put by Lynden Bell in 66 with finding the By the way I knew a personal story I met tearing several times and I have been honored to serve in the committee of the Erwin Schrodinger Institute in Vienna that he created and he was an amazing person so I remember once during dinner for his celebration of maybe he was 80 or something like that he said in physics I tried I tried in physics always to find confirmation of something and I always found a contradiction so also this way of thinking is very interesting so you can work to confirm a basic thing but he was saying that in trying to confirm something you find a contradiction that leads you to another direction so I think it was a very interesting lesson also and he has several examples of this kind in his papers where if you look in detail what the theory that you have you discover aspects that at the first sight you neglected and they are important and this leads you to new directions of investigations unfortunately this paper in 1970 remained a sort of unknown for several years I think we discovered it at the end of the 90s it was a sort of unknown ok, 5 minutes of break then we hope to finish at one so we take 5 minutes break recording stopped if you take the ensemble if you take the ensemble at constant pressure so we start again hello so we start again thank you recording in progress sorry it was a problem with the microphone sorry I forgot the microphone so I repeat there is an extra term which is a repulsion in the volume maybe I am wrong here I don't remember so I have to check the sign but I'm sure it is a repulsion for the case in which we have states and then we ok so so this is the modified theory model in one dimensions in two dimensions and in three dimensions so particles have the sides v1, repel and they attract in the volume v0 so this is more or less the picture so then I compute the canonical partition sum already till indeed and then by Laplace transforming by Laplace transforming I get the unconstrained partition sum so I have to solve this problem here it's not difficult to solve I do it as usual with the Laplace method so and unfortunately there is one aspect it is technical but it is important that the volume I've told you that the volume of tilling is this one you see that it's ok, this is now explicit this is the integral that you see on the blackboard the integral over the positions divided by the Gibbs factor combinatorial factor it's a sum over n0 and n1 with the constraint that n is the sum of n0 and n1 so there is a chronicle delta here and then the tilling in the tilling case there is no sigma so it's v0 to the n0 v1 to the n1 divided n factorial ok but in the case in which there is a sigma in which there is a size for the particles the the expression is correct only in one dimension so there are corrections in dimension 2 and dimension 3 so in fact when we published the paper we were not aware of this difficulty we were not expert enough of question of state for systems of so now we have a paper that we are preparing that corrects this mistake in this paper so ok so this is the the first time you see an expression so this is the replica energy at volume v0 volume v and with a certain number of particles is the explicit expression of the replica energy and here k is a sum over 0 and 1 ok there are 2 containers either you are inside v0 outside v0 and there is an explicit formula for the you can compute then you use the steering approximation then you do the saddle point so large n and you get the replica energy and the replica energy function of the thermodynamic parameters of the system so is like the entropy the free energy is so it would be 0 if the system were additive and clearly this system is not additive because the energy goes like n square so the system is not additive and so you get a finite a finite thermodynamic potential epsilon in order to express this thermodynamic potential in terms of the 3 parameters that now fix the state of the system so the which are pressure chemical potential and temperature you have to solve in this set of consistency equation so you see that they are implicit because you see that temperature appears here for instance so you have to invert you have 3 equations and 3 unknowns so you have to invert in mu p t as a function of the averages because here everything fluctuates so the volume fluctuates so so you have an average volume the number of particles fluctuates so you have an average number of particles both in the inner part and in the outer part and you get them as a function of tp and mu for instance when sigma equals 0 we get these we get these and it's very interesting the form that is taken by by the replica energy is minus the potential plus pv so if you go and look at the thermodynamic relation you see that it is related to the thermodynamic relation that we wrote in the first part of the talk so we have also sort of reasoning where we we take the excess part of this quantity with respect to short range so in some sense there is a long range contribution to the pressure this part of the pressure that come from the gas but what appears in this relation is the pressure due to the long range minus the pressure which is due to the standard short range gas so this you can see in these expressions so it's a long and tedious work to do so we have a long paper about that here I essentially recall some result so equilibrium configuration so for sigma equals zero so when the particle do not have a size like in the theorem model equilibrium configurations exist in the unconstrained ensemble only if b is negative so for repulsive interactions for b equals zero which is the theorem model no equilibrium states exist in the unconstrained ensemble and then there are other functions for instance the unconstrained ensemble are equivalent when b is negative in the grand canonical ensemble equilibrium states exist also for b positive so also when both are positive so when you have both attractions in the two regions and some of these states have negative isothermal compressibility this is one of the response functions that can be negative no phase transition is present in the canonical ensemble the grand canonical ensemble is inequivalent to the canonical ensemble and in this lateral ensemble so in the canonical ensemble phase transition of first order critical point is present in analogy with the theorem model and moreover negative compressibility states appear in the canonical ensemble also for b negative so there are different behaviors in the isothermal now that we have a very complicated set of ensembles we can check several properties and they are different in the different ensembles depending what choice of parameters you do all that for sigma zero sigma non zero as I have said is correct in one dimension so so we introduce a new set of variables which because in this case we have a and b so slightly different from the set of variables that theorem has so adding a feature and then we have so we have a different form of so this is the replica energy it is scaled also for sigma positive and the new feature is that with sigma positive you get phase transitions for instance here is a first order phase transition I am now plotting this function which is unusual free energy so it is a free energy that does not exist for system with short range interactions and here I show that by varying one parameter I have one case in which a minimum is here at these values of x zero which is the fraction of rescaled fraction of particles that are in the box v zero the minimum is here so I have a state here and this is a metastable state and if I change a parameter in this case psi I don't remember one of the rescaled parameters maybe the chemical potential I go from this minimum here to this minimum here so I have a phase transition of first order by varying one parameter and it can be very complicated so you see here the space of parameter is larger so this is the analogous of the opening of the of the forbidden phase in the plane that will not now appear in the space because I have more parameters so it is slightly difficult these are this gray surface there the limits the region in this three-dimensional space where I can have states below on the other side of the gray surface I don't have states there are no equilibrium states on the left side there are equilibrium states and there is a line in the unconstrained ensemble of second order phase transition with two surfaces and the upper surfaces and the lower surfaces the limit region where there are no states so you have jumps if I vary for instance if I vary x bar in the in the vertical direction at some point I have a jump like temperature jumps this type of discontinuities that we have seen in the simpler models it's a richer situation, a richer model this is for instance an example of in equivalence of isothermal isobaric with unconstrained in the isothermal isobaric I have a line transition ending on to a critical point in the case of the unconstrained I have a line I have a spinode line one could call it a line which delimits states that do not exist in this ensemble and finishes on a second-order critical point so I have a very unusual situation and this is something like a cut in the previous three-dimensional plot so I'm cutting a specific region of these are really details so I could give other details of these models and this is very recent so this is the example how you do simulation in this system you need to reservoir one for the particles and one for the volume and then it's simply Monte Carlo acceptance and probabilities for configurations so you have to move the particles around and then compute the acceptance rate according to how you vary the energy and here the energy is more complicated so the energy is the replica energy so you have to count in a new way but is a standard Monte Carlo so now for instance here is a simulation where we don't put the inner volume V0 doesn't exist particles only repel and they stay in volume V0 so particles stay in volume V0 and they only repel and there are equilibrium states and we reach this equilibrium state with Monte Carlo very precise do you see so these are the lines that these models can be easily solved one is one in the class of theory models and it's easily solved and then we compare the analytical solution with numerical simulation and this is a rather different case in which now uncomparing isothermal and isobaric there is a side for the particles now and there is repulsion and you see that you have a typical situation of inequivalence so in one ensemble I follow the line and I reach metastable states and then I have a region of inequivalence because this quantity cannot increase in one ensemble while in the unconstrained I have a first order first order phase transition in the unconstrained the maximum construction that you have seen in the in the first lectures so as usual when you go in an uncharted region you try to explore so we are continuing the exploration of these different models in this uncharted region of the theory model but of course one could try also with other models and also more realistic models so in these lectures I have tried to convey the idea that in long range interactions we have a new ensemble which can have equilibrium states phase transitions like the other ensembles we are calling this ensemble the unconstrained ensemble the terminal potential for this unconstrained ensemble is the replica energy the replica energy can be computed explicitly even analytically in some models I have not shown but it can be computed also for one over r to the alpha interactions in the mean field approximation and the unconstrained ensemble can be equivalent to the other ensembles but cannot also so there are cases of inequivalence and ok, this is just an exploration that is going on we are now using equations of state for fluids in two and three dimensions to correct the the solution that we have published which is valid only in one dimension and that's all I think since this is my last lecture I think you can leave it open for discussion and if you have also questions to put on previous previous lectures starting from tomorrow Nikolaj Defeno will give lecture on the quantum quantum mean field is in model ok questions? don't be shy anyhow you will be here I will be here tomorrow and I'm available for you will do the teaching assistant I will do the teaching assistant I will try to reproduce the calculation of Nikolaj because some of them I don't fully control because I'm not so I don't know much about quantum systems but I wanted to give a flavor of with these three more lectures all these things are being now exported to the quantum domain finding fish that are similar fish that are new so it's interesting ok, so I think we can close here and have lunch so, thank you again